Working Paper 02-33 Economics Series 14 February 2003

Departamento de EconomíaUniversidad Carlos III de Madrid

Calle Madrid, 12628903 Getafe (Spain)

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ASSESSING THE EFFECTOS OF MEASUREMENT ERRORS ON THE ESTIMATION OF THE PRODUCTION FUNCTION *

Carmine Ornaghi1

Abstract This article explores the reasons why GMM estimators of production function parameters

are generally found to produce unsatisfactory results. I attribute this finding to the inaccurate construction of the variables used in production function analysis. In particular, I suggest that the problem lies in the use of common price deflators as well as in a capital variable that does not reflect the true flow of capital services because of short-run equilibrium effects. I show that the practice of using industry-wide deflators leads to lower scale estimates, mainly due to a relevant downward bias in the labour coefficient. At the same time, it introduces a large upward bias in estimating the elasticity of output with respect to technological innovation. Moreover, a significant improvement in the estimates of capital coefficients is obtained if the information on the degree of capacity utilization is adequately exploited. Keywords: Scale economies; Price deflators; Temporary Equilibrium. JEL Classification: C23, L6.

1 Departamento de Economía, Universidad Carlos III de Madrid. E.mail: carmine@eco.uc3m.es * I wish to thank Jordi Jaumandreu and Pedro Marin for many helpful discussions and suggestions.

1 Introduction

Despite the wide literature available dealing with empirical estimation ofproduction functions, there are some interesting and challenging questionsthat have not been solved. Although simple OLS regression gives plausibleparameter estimates, consistent with constant return to scale, econometri-cians agree that the usual exogeneity assumptions of the regressors that arerequired for the consistency of OLS are unlikely to hold. To the extent thata productivity shock is anticipated before the optimal quantity of inputs ischosen, the production function disturbances are transmitted to the inputdemand equation. This can lead to an estimation bias due to the correla-tion between the right-hand variables and the error term (often defined as“simultaneity bias”). Moreover, it is likely that each firm is characterised byspecific factors of production, such as entrepreneurial ability, that are notobservable but that can affect the current input choice, thus worsening thebias in OLS estimates.

The availability of company panel data provides the necessary set of re-sponses to get around the strict exogeneity requirements of the regressors.As long as one is willing to assume that productivity differences among firmstend to be rather persistent over time, one can proceed to first difference thepanel data to eliminate firm-specific effects. At the same time, panel dataprovides the necessary instruments to correct for simultaneity in this firstdifferenced equations. The most widely used technique is the GeneralizedMethod of Moment (GMM ) that relies on combining in an optimal way a setof orthogonal conditions generally defined using suitably lagged levels of theinputs as instruments.1

In empirical practice, attempts to control for unobserved heterogeneityand simultaneity generally give less satisfactory results: low and often in-significant capital coefficients and unreasonably low estimates of returns toscale. A number of reasons has been put forward to explain the endemicproblems found with estimators in differences. Blundell and Bond (2000)suggest that these problems derive from the weak correlation that exists be-tween the growth rates of capital and employment and the lagged levels ofthese inputs. Being the capital and the employment series highly persistent,

1See Arellano and Bond (1991) for detailed discussion.

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past levels of these variables contain little information about the specificationin first differences. They show that estimation can be notably improved byadding lagged first differences as instruments for the equation in levels. Usingthis extended GMM estimator, they find a higher and significant coefficientfor the capital and they fail to reject the constant return to scale hypothesis.The explanation and proposed solution both lie in the econometric field.

On the contrary, Klette and Griliches (1996) - hereafter KG - suggestthat the implausible low estimates of returns to scale can be explained bythe incorrect measure of the output. They affirm that “the practise of usingdeflated sales as a proxy for real output will, ceteris paribus, tend to createa downward bias in the scale estimate obtained from the production func-tion regressions” (pag. 344). They stress that the estimated parameters inproduction function analysis are reduced form parameters that derive fromthe interaction of the production function with the demand equation. Trueelasticities of scale can only be inferred modelling a more general frameworkwhere the production function is augmented with a demand equation and aprice setting rule. The authors present also an empirical estimation to illus-trate their theoretical approach. Unfortunately, without the availability offirm level data, it is impossible to know the magnitude of the bias in scaleestimate due to the use of common price deflators and we can not contrastwhether reinterpreting the parameters as reduced form coefficients actuallyprovides a reliable solution to this bias.

The first aim of this paper is to provide interesting evidences on thismatter. The data base used reports variations in output prices at the firmlevel. This allow us to construct two measures of output: first deflating salesusing firm-level price data (the correct measure of output) and second usingindustry-wide deflators (the incorrect but generally used measure of output).Apart from investigating the impact on input coefficients for a standard pro-duction function, I explore the consequences of using deflated revenues whenestimating price-cost margins and scale economies. The results obtained sup-port the perspective of KG. I find that scale estimates and price-cost marginsare downward biased when using deflated sales (or deflated value added) asdependent variable in the production function equation.

Additionally, evidences are provided on two aspects that have been largelyunnoticed in this literature. First, the fact that the difficulty pointed out for

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deflated sales needs to be extended to intermediate consumptions. Giventhat most of the available data set do not report a “quantity” measure ofmaterial, the general approach is to deflate the normal expenditure in inter-mediate inputs by a common (and often unique) deflator. The underlyinghypothesis made is that there is perfect competition in this factor market,so that all the firms are charged with the same price. However, significantdispersion in input prices seems to exist between firms in the same industry.2

The theoretical part of the paper shows that the use of deflated expendituresin materials exacerbates the bias in the point estimates of labour (already)introduced by the use of deflated sales but it can potentially offset the biasin estimating the coefficient for materials. Using the available data on in-termediate input prices for individual firms, I present evidences supportingthis result. Second, the consequences of using incorrect measure of outputand intermediate inputs on estimating R&D output elasticities. Since theseminal paper of Griliches (1979), many authors have investigated the pro-ductivity effects of R&D (generally defined as knowledge capital). I find thatthe practice of using common deflators tend to create a large upward bias inthe estimation of the R&D coefficient.

Another reason that has been advanced to justify the poor results ob-tained in the first difference GMM framework is the lack of information onquality of labour and capital and, even more important, on capacity utilisa-tion. As long as demand shocks or sudden changes in factor prices, such asthe energy price shocks of 1973 and 1979, lead to under or over-utilisationof capacity, producers may find themselves in short-run or temporary equi-librium. As suggested by Berndt and Fuss (1986) (hereafter BF), in thesecircumstances the growth rate of output can hardly be explained by the vari-ation of capital as reported in standard company data set. It can be thecase that increases in the output produced in two subsequent periods can beaccommodated through changes in the utilisation rate of equipment withoutany further investment in capital. The existence of a temporary equilibriumrequires the adjustment of the (quasi-fixed) capital variable in order to get a

2Differences in the prices of materials paid by manufacturers can be due to differencesin firms’ purchasing power or because of incomplete information on the prices charged byall potential suppliers. Large firms, for example, are likely to get lower prices becausethey can afford higher searching costs (when chosing among suppliers) and because ofthe discounts that are usually granted on large buying. Beaulieu and Mattey (1999) findevidences of large dispertion in the prices of materials for same US industries.

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correct measure of the capital coefficient in production function regressions.

The second goal of this paper is to explore the possibility of improvingthe estimation of the capital coefficient by taking into account the short-runrelative factor usage. Although the relevance of these temporary equilibriumeffects has long been recognised in productivity analysis and growth theory,this issue has been largely ignored in empirical research, mainly because ofthe lack of suitable micro data. I show that the use of capacity utilisationdata at the firm level can significantly improve the estimation of the capitalcoefficient, avoiding the low estimates that are usually found in most of thestudies.

The empirical part is based on a micro panel data set of Spanish manufac-turing firms that includes about 2000 entities during the period 1990-1999.In addition to the standard data on firms’ sales and factors of production,there is access to output and intermediate input prices as well as the averagepercentage of capacity used by the firm along the year. This data set allowsme to compute a “true” measure of output and material and to adjust thestock of capital for short-run equilibrium.

The rest of the paper is organised as follows: Section 2 provides a generaltheoretical analysis of the bias in the production function regression whenusing deflated sales and deflated expenditure in intermediate inputs. Theanalysis is extended to the estimation of capital coefficients considering theinformation on utilisation of capacity. Section 3 describes the data and ex-plains how the variables are constructed. Section 4 discusses the empiricalresults while Section 5 draws some final comments.

2 Theoretical and Econometric Framework

In this section I provide a set of theoretical underpinnings to clarify the con-sequences of using incorrect measure of output and factors of production.KG show that if real but unobserved prices are correlated with the explana-tory variables included in the model, an omitted variable bias will arise. Thesame analysis needs to be extended to intermediate input prices: using com-mon deflators for expenditures in materials can exacerbate or reduce the bias

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mentioned above. Moreover, plausible assumptions suggest that the R&D co-efficient is upward biased. In the second part of the section, I discuss theimportance of modifying the standard approach used to estimate the capitalcoefficient in order to accommodate forms of temporary equilibrium.

2.1 Deflators Bias

To keep things as simple as possible, assume that the production functionfor manufacturing firms can be represented by a Cobb-Douglas function inthree “conventional” inputs,3 labour L, physical capital C and materials M ,augmented with a technology parameter based on the firm research effort R:

Qit = Lα1it M

α2it C

α3it R

α4it e

upit (1)

where Q is the quantity produced and up is the random error term forthe equation, representing the effect of efficiency differences, functional formdiscrepancies and measurement errors. Subscripts are reported only when itis strictly necessary to avoid confusion.

Taking logarithms and first differences to eliminate fixed effects that areburied in the residual, we obtain the linear equation:

qit = α1lit + α2mit + α3cit + α4rit + upit (2)

Where lowercase letter with a tilde stands for logarithm differences be-tween year t− 1 and t.Assume that deflated sales is the available proxy for real output. Then,

the relationship between output growth, q, and deflated sales, y, is:

qit + pit − pIt = yit (3)

where pit is the growth in firm i ’s specific price while pIt is the growth inthe industry-wide deflator.4

3As stressed by Griliches and Mairesse (1984, pag. 342), “one could, of course, considermore complicated functional forms, such as the CES or Translog functions. [..] this willnot matter as far as our main purpose of estimating the output elasticities of R&D andphysical capital, or at least their relative importance, is concerned.

4Suppose that we have data on sales, Pit ∗ Qit and we use a common price deflatorsPIt to get deflated sales Yit = (Pit ∗Qit)/PIt. Taking logarithms and first differences, weobtain yit = qit + pit − pIt.

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Substituting (3) in equation (2), we get:

yit = α1lit + α2mit + α3cit + α4rit + (pit − pIt) + upit (4)

The latter equation shows that when deflated sales, Y , replace the truemeasure of output, Q, the omitted price variable (pit − pIt) enters into theresidual. As long as the optimal choice of factors of production is affected bychanges in firm prices, the omitted price term tends to create a bias in OLSestimator of the parameter vectors α.5

At the same time, if the variable that we observe is not real materials,M , but deflated expenditures in materials, N , then we have that nit = mit+git − gIt, where git and gIt are intermediate input prices at firm level andaggregate level, respectively. Accordingly, equation (4) should be restated asfollows:

yit = α1lit + α2nit + α3cit + α4rit + (pit − pIt)− α2(git − gIt) + upit (5)

The term (git − gIt) can possibly exacerbate or offset the biased intro-duced by the output price variable (pit− pIt).6 The difficulties outlined abovecannot be solved by using appropriate instrumental variables. Potentiallyuseful instruments, which have to be correlated with the inputs growth, arelikely to be correlated with the input and output price changes buried inthe residual and therefore are illegitimate instruments. Our results, obtainedusing instrumental variable method confirm this perspective.

Following the analysis presented in KG, the consequences of using com-mon deflators on estimating the production function parameters can be il-lustrated defining a log-linear demand equation for firm i of the followingtype:

5Let us define the term (pit − pIt) as Πit. As shown in Klette and Griliches (1996),pag. 346, the bias in the OLS estimator of the α coefficients depend on the sign and themagnitude of the parameter δ in the auxiliary regression: Π = Xδ + uΠ where X is thematrix of the growth in inputs and uΠ is a white-noise error term.

6Let us define the term (git − gIt) as Ωit. Using a symmetric argument to the one inthe previous footnote, we have that the OLS estimator of the parameter vector α dependsalso on the auxiliary regression: Ω = Xγ+ uΩ. Then, the overall direction and size of thebias depend on the sign and magnitude of δ and γ.

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Qit = QIt ∗ (Pit/PIt)ηRβ1it e

udit (6)

where Qit stands for firm i ’s sales, QIt is the industry output, Pit isthe firm specific price while PIt is the average price in the industry. Rit isthe knowledge capital of the firm measuring the innovative content of itsproduct. This variable acts as a demand shifter: increasing the quality ofthe product through innovation boasts the sales of the firm.7 Rewriting thedemand system in terms of growth rates (more precisely as first differencesof logarithms), we obtain:

qit = qIt + η(pit − pIt) + β1rit + udit (7)

Now, using equation (3) and rearranging, it follows that:

(pit − pIt) = 1

1 + η(yit − qIt − β1rit − udit) (8)

Equation (8) gives an analytical interpretation of the (usually) unobservedterm (pit − pIt). Apart from the residual, the only variable that is unknownis the growth in industry output qIt. As shown by KG, it is possible to usethe weighted average growth in deflated sales of all the firms in the sampleas a proxy for this term . We will come back to this point in Section 3.

Combining equation (5) and (8), we get:

yit =η + 1

η(α1lit + α2nit + α3cit) + (

η + 1

ηα4 − β1

η)rit

− 1ηqIt − α2

η + 1

η(git − gIt) + vdit (9)

where v is an error term that captures both demand and productivityshocks.

Equation (9) shows that the estimated parameters for the productionfunction have to be interpreted as reduced form parameters (coming up fromsupply and demand coefficients). Under a variety of situations, the omittedprice variable (pit− pIt) tends to create a downward bias in scale estimate for

7See Klette and Griliches (1996) and Klette (1996) for further details.

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OLS technique.8 As we show in the empirical part, the growth in industryoutput qIt plays a crucial role to identify the true coefficients for the standardinputs of production.

Equation (9) differs from the analysis of KG because of two terms: theknowledge capital, rit, and the omitted material price, (git − gIt). Noticethat the two terms making up the coefficient of rit capture the effect of bothproduct and process innovation. The coefficient (η+1

ηα4 − 1

ηβ1) in equation

(9) is greater than α4 in equation (2) if β1 is larger than α4.9 In a companionpaper, Ornaghi (2002), evidence supporting this assumption is provided.10

We can then infer that the use of common price deflators tend to create anupward bias in the estimation of the productivity effects of R&D. The resultspresented in Section 4 suggest that this upward bias can be rather large formanufacturing firms: using deflated sales as dependent variable can lead toestimate an R&D coefficient that is almost twice as large as the one we obtainusing the “true” output measure.

As far as the term (git−gIt) is concerned, it seems plausible that firms thatexperience higher costs of materials will, ceteris paribus, reduced the quantityused of this input in favour of the other short-run factor of production, such aslabour.11 In econometric terms, this is equivalent to assume the existence ofa systematic negative correlation of the generally unobserved term (git− gIt)with materials and a positive correlation with labour. Given the negativesign in front of the (git − gIt) coefficient, the downward bias in the estimateof input elasticities due to the omitted output price (pit− pIt) is exacerbatedin the case of labour but it is (partially or fully) offset for materials.12 Theestimates shown in Section 4 confirm that the coefficient of materials under

8The term (pit − pIt) is a source of downward bias for OLS estiamates whenever it isnegatively correlated with the explanatory variables in the equation. This seems likely ifthe firm market power arises mainly from a specific demand for its products. Detaileddiscussion of this issue can be found in Griliches and Mairesse (1995).

9Take the coefficient (η+1η α4 − 1ηβ1) and rewrite β1 in terms of α4 as β1 = nα4, with

n ∈ R+. Rearrenging, we obtain that 1η (η − (n − 1))α4. This term is greater than α4 ifn > 1, that is if β1 > α4. Moreover, the greater is n, the higher is the coefficient of r it inequation (9) compared to the one in (2).

10See Garcia et all. (2002) for a similar result.11The assumption made is that these two inputs are to a certain extent substitutes in

the production process.12Suppose that we have a correct measure of output, Q. Moreover, we know total ex-

penditure in intermediate inputs but we do not have individual prices charged to the firm

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equations (9) - that is, when general deflators are used - and (2) - when the“true” measures of output and materials are used -are not sensibly differentfrom each other, but the labour coefficient is seriously biased downwards.

2.2 Temporary Equilibrium

Awell known finding in productivity analysis is the low and often insignificantestimate of capital coefficients when the production function is specified indifferences. My claim is that this persistent puzzle is essentially due to theexistence of temporary equilibria characterised by under or over utilisationof firms’ installed capacity. If the production function is stated in levels,it seems plausible that there is always a positive and significant correlationbetween the firm output and its available total stock of capital, even whenproducers are not in a long-run equilibrium. Things are different when weconsider a regression in first differences. Increases in the output producedin two subsequent periods can now be accommodated through variations incapacity utilisation of equipment and machinery, without undertaking anyinvestment. Appendix C presents a simple econometric model to show thatmeasurement errors are more likely to severely affects the estimation of thecapital coefficient when we go from levels to differences.

The relevance of temporary equilibria, especially those associated withbusiness cycles, has long been recognised in productivity analysis. In theirarticle dated 1986, BF suggest to adjust the quantities of the quasi fixed fac-tors for temporary equilibrium effect. The theoretical underpinnings to this

for these inputs; this means that we can construct a proxy for material using generaldeflators. Then, the (first-differenced) specification of our production function would be:qit = γ1lit + γ2mit + γ3cit + γ4rit + γ5(git − gIt) + uit Let define (git − gIt) as Ωit. TheOLS estimate of the labour and material coefficients would be:γ1 =

cov(lit,qit)

var(lit)− γ5

cov(Ω,lit)

var(lit)= α1 − γ5

cov(Ω,lit)

var(lit)

γ2 =cov(mit,qit)var(mit)

− γ5cov(Ω,mit)var(mit)

= α2 − γ5cov(Ω,mit)var(mit)

where α1 and α2 are the true coefficients. It is likely that firms that experience a highergrowth in prices of intermediate inputs, Ωit, will try to substitute this input with labour.For example, firms can decide to reduce the “outsourcing” of their activities when thesecontracts became so expensive that the cost of performing it inside is less than contractingit out. This reasoning implies a positive value of the covariance term in the first equationreported above and a negative value for the corresponding term in the second equation.Then, we can expect that the use of general deflators leads to γ1 < α1 and γ2 > α2.

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adjustment rely on the fact that the stock of capital should always reflect theunobservable available services from this input. However, the lack of microdata has hampered the possibility of defining adjustment procedures. In thispaper I define two variants of the production function equation (2) above inorder to deal with temporary equilibrium effects. Detailed explanations ofthese procedures are postponed to the next Subsection. Results presented inSection 4 confirm that estimates of output elasticities with respect to capitalare notably improved by exploiting the information on capacity utilisationrates.

2.3 Input Elasticities, Scale Eonomies and Mark-ups

In this section I state the estimating equations considered in the empiricalanalysis. All the models are estimated using, on one hand, a “quantity”measure of output, Q, and materials, M (that is, the true but usually notobservable measures) and, on the other hand, deflated sales, Y, and deflatedexpenditure in intermediate inputs, N (the variables generally used in mostof the empirical papers). All these models provide interesting evidences onthe extent of the bias due to the incorrect measurement of the variables. Inthe first part, the capital input C correspond to the total stock of equipmentas reported by the firm and a measure of capacity utilisation, CU , is addedas a (independent) regressor in the production function. In the second part,I provide the alternative specifications dealing with short-run equilibriumeffects.13

Consider equation (2) above. Under constant return to scale with respectto standard factors of production (labour, materials and physical capital),we have that α1 + α2 + α3 = ε = 1. For interpretive reasons, equation (2) isrestated so that deviations from constant returns are measured explicitly:

qit− cit = α1(lit− cit)+α2(mit− cit)+(εob−1)cit+α4rit+α5fcuit+ uobit (S1)The corresponding specification using deflated variables is:

13I prefer to start with a more “conservative” specification based on a standard definitionof the capital input, C, in order to clearly disentangle the “omitted price” bias from theeffects of temporary equilibrium.

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yit− cit = γ1(lit− cit)+γ2(nit− cit)+(εunob−1)cit+γ4rit+γ5fcuit+uunobit (S2)

Notice that εob and εunob correspond to the scale elasticity of standardinputs when “quantity” measure of output and materials are observable orunobservable, respectively. Moreover, the error term in (S2) picks up dif-ferences between changes in firm output and intermediate input prices andthe corresponding industry price indexes, (pit − pIt) and (git − gIt). Section4 shows estimation results when value added is used as dependent variable,too.

Equation (9) above shows that in principle the growth in industry out-put, qIt, ensures identification of demand elasticities and production functionparameters. If this were the case, true input elasticities obtained from es-timating equation (S1) can be inferred, even when we use deflated grossproduction and intermediate input expenditure, by adding the term qIt tomodel (S2), as follows:

yit − cit = γ1(lit − cit) + γ2(nit − cit) + (εunob − 1)cit+γ4rit + γ5fcuit + γ6qIt + u

unobit (S3)

Results presented in Section 4 are partially consistent with this approach,first modelled by KG.

I have also considered a different specification of the models above. Fol-lowing the works of Hall (1988) and others,14 the elasticities of short-runinputs, labour and materials, can be replaced by their income shares cor-rected for the presence of market power:

α1 = µ ∗ ωl and α2 = µ ∗ ωm (10)

where µ is the ratio between price and marginal costs, ωl and ωm are the(observable) income shares of labour, ωl = (CostLabour/Sales), and mate-rials, ωm = (CostMaterials/Sales).15 Under these conditions, the resultingestimating equations are:

14See, for example, Klette (1999) .15On an empirical ground, the shares of labour and materials costs in total revenues, ωl

and ωm, are computed as the averages over adjacent years (i.e., following the Tornquistapproximation).

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qit − cit = µob1 ωl(lit − cit) + µob2 ωm(mit − cit) + (εob − 1)cit+α4rit + α5fcuit + uobit (S4)

and

yit − cit = µunob1 ωl(lit − cit) + µunob2 ωm(nit − cit) + (εunob − 1)cit+γ4rit + γ5fcuit + uunobit (S5)

These later models allow us to determine the extent of the bias in estimat-ing the mark-up coefficient, µ, as well as the scale parameters, ε, when truemeasures of the variables, as stated in model (S4), are replaced with variablescomputed with industry-wide deflators, as in model (S5). Although theoret-ical models predict the existence of a unique markup term (see equation (10)above), we prefer to estimate µ1 and µ2 separately in order to see their vari-ation across specifications. This approach can be used as a simple test toassess the effects of industry-wide deflators: as long as the assumption that(git−gIt) intensify the downward bias of the labour coefficient while softeningthe bias for material holds, we should find a discrepancy between µunob1 andµunob2 . If this is the case, µunob2 should be considered a better indicator of themarket power exercised by firms in an industry. I come back to this point inSection 4, after presenting the results for specification (S4) and (S5).

As for specification (S3), the variable qIt can be added to (S5) in orderto retrieve true mark-up factors when industry-wide deflators are used:

yit − cit = µunob1 ωl(lit − cit) + µunob2 ωm(nit − cit) + (εunob − 1)cit+γ4rit + γ5fcuit + γ6qIt + u

unobit (S6)

Finally, I present the estimating equations when allowing for the existenceof temporary equilibria. As advanced in Section 2.2, two different proceduresare defined in order to account for these short-run effects. First, I separatethe estimates of the capital coefficient depending on the degree of capacityutilisation reported by the firm:

qit = α1lit + α2mit + αh3D1cit + αl3D2cit + α4rit + α5fcuit + upit (S7)

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where D1 is a dummy that takes value 1 if the capacity utilisation in bothyear t and t+1 is greater than 80% while D2 takes value 1 if this conditionis not satisfied.16 We expect the parameter αh3 to be a better estimate of“true” capital coefficient as the proportionality assumption used to identifycapital flows from stocks does not hold when factor usage is rather low.17 Asa second approach to the estimation of a more reliable capital coefficient, Idefine a “short-run” capital input, C*, by multiplying C and CU. Then weestimate the equation:

qit = α1lit + α2mit + α3cit + α4rit + upit (S8)

using C* as instrument of the quasi-fixed factor C.18 The econometricunderpinnings of specification (S8) is found in the context of errors in vari-ables. The capital stock C is an erroneous measure of true capital serviceswhen capacity utilisation is not full.. I then use C* as instrument to softenthe measurement error. Notice that the last two specifications are defined soas to get a direct estimate of the capital coefficient, α3, and not to determinedeviation from constant return to scale, (ε− 1), as in previous equations.

2.4 Other Econometric Issues

As mentioned in the introductory section, the required predeterminacy of theright-hand variables to get consistent OLS estimates is unlikely to hold for

16As explained in Appendix C, the downward bias in the estimate of the capital coef-ficient is minimized when we use high values of capacity utilization for two consecutiveyears. I opt for a threshold value of 80% since it is the average capacity utilisation re-ported by the firm as shown in Appendix A. Results are robust to alternative specifications(higher than 80%) of this value althought coefficients are less precisely estimated.

17On this point BK (pag.11) affirm that “when economic utilization is full, flows fromquasi-fixed inputs can be assumed to be proportional to the stocks [...]. This leads to thereplacement of unobserved service flows [...] by observed stocks”

18I could alternatively use C* as regressor in equation (S8) but I think that specification(S8) allows to differenciate those firms that use the same amount of capital services butwhose stock of capital is different. For example, suppose that two competing firms areusing two equipments each but one of these firms is not using a third equipment. In thiscontext, the stock of capital, C, of the latter is greater but because of under utilisation, theshort-run capital services, C*, that the two firms are actually using coincide. The capitalinputs must be differentiated because one firm has capital services readly available withoutaffording any installation cost. Moreover, the capacity utilisation is an average rate forthe entire year and it can possibly not reflect the true factore usage in some periods.

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some inputs of the production function. In particular, I assume that short-run factors of production, labour and materials, are possibly correlated withthe error term. To avoid the “simultaneity bias” due to the transmission ofproductivity shocks, I use GMM estimators with lags of employment andmaterials as instruments of the endogenous variables. As far as the capacityutilisation is concerned, I test for the endogeneity of this variable in eachspecification using the “Incremental” Sargan Test.19 I fail to reject the nullhypothesis of strict exogeneity only for models (S4), (S5) and (S6). As shownin Section 4, lagged levels of CU are used as instruments when estimatingall the other specifications.

Additionally, anecdotal evidence suggests that firms do not adopt a uni-form criterion when reporting the R&D expenditures incurred during theyear. Therefore, it is likely that the R&D stock (see Section 3 below) is sub-ject to some measurement errors, creating a problem of endogeneity similarto the one due to simultaneity. Again, this problem is tackled using lags ofthis variable as instruments. In Section 4 I report the exact set of instrumentsused to estimate the alternative specifications stated above.

3 Data and Variables

The data used in this study are retrieved from the Encuesta sobre Estrate-gias Empresariales, ESEE, (Business Strategy Survey) an unbalanced panelsample of Spanish manufacturing firms published by the Fundación EmpresaPública covering the period 1990-1999. The raw dataset consists of 3,151firms for a total number of 18,680 observations. A “clean” sample is definedaccording to a set of criteria which are given in Appendix A. The sample em-ployed here consists of all the firms that have been surveyed for at least threeyears after dropping all the time observations for which the data required tothe estimation are not available. Firms are classified in 15 different sectors,which are listed in Appendix A. This Appendix also reports an explanation

19This test is defined using the Sargan Test, S, and the relative degrees of freedom, (df),of two separate regressions: one considering capacity utilisation as an exogenous variable,S1(df1), and the other as an exogenous variable, S2(df2).The difference between the twoSargan Test, S1−S2, is distributed as a χ2 with degrees of freedom (df1 − df2).

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of the variables used across the specifications stated in Section 2.3 togetherwith descriptive statistics.20

A unique feature of the data set is the availability of information on theprice changes set by the firm together with price changes charged to the firmfor its intermediate inputs. This allows me to construct a proper measure ofoutput, Q, and materials, M .21 Subtracting the latter from the first, we alsoobtain a quantity measure of value added, V A.

Complementary information about price deflators for gross production,expenditures in materials and gross value added are taken from the (Spanish)National Institute of Statistics (Instituto Nacional de Estadistica - INE).The series of output deflators for gross production at fifteen-industry levelcover all the period 1990-1999 while those for value added are limited to theperiod 1990-1997. Using these series, two alternative dependent variablesare defined: deflated gross production, Y , and deflated value added, deV A.Deflated materials, N, are computed using an overall deflator for non-durablegoods. These alternative measures of output and intermediate inputs allowus to infer the magnitude of the bias due to the use of industry-wide deflators.

Labour inputs are represented by man hours, taking into account overtimeand lost hours.22 It is likely that this variable is affected by measurementerrors due to rounding-off. The number of employees, EMP, is then used asinstruments of the hours of work when estimating the production function.23

Capital, C, is computed recursively from an initial estimate (based on bookvalues adjusted to take account of replacement values) and data on firms’investments in equipment goods:

Cit = (1− δj) ∗ Cit−1 + Iit (11)

20Further detail on the dataset can be found in Garcia et all. (2002) and Ornaghi (2002).21Suppose that we have data on gross production (at seller price) for two consecutive

years: Pit ∗Qit and Pit+1 ∗Qit+1. Having access to data on price changes, we can expressthe figures above in terms of the reference year t: Pit ∗Qit and Pit ∗Qit+1. At this pointif we take log first-difference, we have a measure of the output growth rate, (log(Qit+1)−log(Qit)), free from price effects.22More precisely, this variable has been constructed using the number of workers ad-

justed for the “double counting” of R&D employees, times the normal hours plus overtimeand minus lost hours.23This is more a formal than a substantial problem. Estimation results obtained using

lag valued of L instead of EMP gives similar results.

16

The subscript j states that we use sectorial estimates of the rate of depre-ciation, δ. Real capital is then obtained using an overall investment deflator(for durable goods). Another interesting feature of this survey is the avail-ability of data on capacity utilisation, CU . As mentioned above, I constructan alternative measure of capital, C∗, to be used as instrument of C in spec-ification (S8). This (short-term) capital input is computed multiplying theoriginal capital input, C, by the reported capacity utilisation, CU.

The R&D capital variable has been computed using standard historicalor perpetual inventory method:

Rit = (1− δ) ∗Rit−1 + Eit−1 (12)

The knowledge capital in period t depends on the previous period stockof knowledge, suitably depreciated at rate δ, plus investments in R&D ofperiod t-1. These investments, E, take into account not only the cost ofintramural activities but also payments for outside contracts and importedtechnology. Equation (11) has been implemented using a depreciation rateof 15%.24 Given that our data are limited to the period 1990-1999, I need todefine a plausible value of the knowledge stock for 1990 before applying thelaw of motion (11). The pre-sample capital is defined using the data on thehistory of R&D expenditures during the eighties and nineties provided by theINE for our 15 industries. Once computed the firms’ average expendituresduring the period 1990-1999 and the associated expenditures at industrylevel using our survey, the individual R&D is assumed to follow the sameevolution of total industry investments for each year since the firm has beenestablished;25 if the firm has been established before 1980, we just considerthe expenditure of the eighties..26

24This is the depreciation rate most frequently used in this type of estimation. In anotherpaper, Ornaghi (2002), I show that estimates of R&D coefficients are robust to alternativespecification of the depreciation rate. Hall and Mairesse (1995) obtain similar evidences.25For example, suppose that firm-i average expenditures during the period 1990-1999

amounts to 5% of total industry-j average expenditures for the same period. Firm-i invest-ments for the previous decade are computed applying this percentage to total industry-jR&D investments as reported by the INE.26The definition of the initial capital does not seem to be a relevant issue in the case

of Spain, considering that the level of R&D investments during the seventies is negligibleand the expenditures during the eighties are sensibly lower than those of the followingdecade. Total R&D investments amount to 46,862 millions of pesetas in 1982 comparedto 246,239 in 1990. The average amount of R&D expenditures for the period 1982-1989 is

17

Finally, the growth rate in industry output from year t-1 to year t, qIt,is estimated using a weighted average of growth in deflated sales for all thefirms in the industry, with the average output shares in the two years asweights.27

4 Regression Results

Table 1 presents the results obtained estimating the production function (S1)and (S2) as well as model (S3) where industry output is added as an extraregressor. The first interesting evidence that obtained comparing columns 1and 2 is that scale elasticities are lower when we use deflated sales, Y, andexpenditure in intermediate inputs, N, instead of output, Q, and materials,M. In particular, the null hypothesis of constant return to scale is rejected at5% significant level only in specification (S2). This result is consistent withthe idea of downward bias in scale estimates advanced by KG. However, look-ing at the estimated coefficients of short-run inputs, it appears that only thecoefficient of labour is downward biased (and largely so) while the coefficientof materials is quite stable across the two specifications. In Section 2, I sup-port the perspective that the downward bias due to the omitted output pricecan be either exacerbated or offset when there is also an omitted input priceburied in the residuals. This result confirms that the practice of using com-mon price deflators do not affect all the inputs coefficients in the same way.Further detail on this point can be found in Appendix B. There is anotherproblem that has been ignored in production function analysis. As suggestedby the theoretical framework in Section 2, the impact of knowledge capitalon firms’ productivity can be overestimated when firm-level price data arenot available. The output elasticity with respect to R&D is more than 70%higher in column 2 than in column 1.

INSERT TABLE 1 ABOUT HERE

All these findings are confirmed when we use value added as the dependentvariable. In column 4 the “true” measure of value added, V A,28 is used while

99,370 millions compared to a higher 262,460 for 1990-1992.27This is the same approach used by KG.28This is defined as the difference between “true” output, Q, and materials, M .

18

in column 5 the dependent variable, deV A, is computed using industry-widedeflators.29 Scale elasticities are more than ten percentage points lower incolumn 5 and, once again, the null hypothesis of constant return to scale isrejected only for this specification. Labour and R&D coefficients are seriouslydownward and upward biased, respectively. Column 3 shows the results ofestimating equation (S3). The coefficient of growth in industry output ishighly significant. Interestingly, adding this industry variable to the modeldo not affect the estimated coefficients for the other variables. As argued inSection 2, true elasticities of output with respect to the factors of productioncan be identified using the coefficient of this new term. Simple calculationshows that model (S3) implies a demand elasticity, η, of -4.31. If we adjustthe labour coefficient (the only parameter that is largely affected by pricedeflators) in column 3 for the term η

1+η(see equation (9) above), we get a

new estimate around 0.38, similar to the result reported in column 1.

Table 2 shows the results obtained estimating models (S4), (S5) and (S6).The same findings stressed above are confirmed under these specifications.Scale elasticities are lower when we use price deflators so that the null hypoth-esis of constant returns to scale have to be rejected. The R&D coefficient istwice as large as the one reported in column 1, confirming a relevant upwardbias in the estimation of the productivity effect of technological innovation.The mark-up parameters µob1 and µob2 suggest the existence of a moderatemarket power across the industries. Interesting enough, the null hypothesisthat these two parameters are not statistically different cannot be rejected.On the contrary, estimates reported in column 2 reveal a relevant downwardbias in the mark-up associated to labour, µunob1 , while the term, µunob2 , doesnot show the same pathology.30 These results show that the separate esti-mation of µunob1 and µunob2 can be considered a sort of test to assess the effectof industry-wide deflators on the mark-up estimates. Moreover, they seemto support the claim that the mark-up coefficient for deflated intermediateconsumption, µunob2 , is more reliable that µunob1 as a measure of market power.

INSERT TABLE 2 ABOUT HERE29As mentioned in Section 3, I find value added deflators only for the period 1990-1997.30Point estimate of µ2 is a little bit higher in column 2 than in column 1 as for the

models presented in Table 1.

19

As far as model (S6) is concerned, we can infer that the demand elasticity,η, is around -6.06.31 If we adjust the estimate of µunob1 in column 3 for theterm η

1+η, we get a new price-cost margin ratio associated to the labour

coefficient of 1.10, very close to the result obtained for specification (S4). Thisresult supports the perspective of KG that input coefficients, scale elasticitiesand price-cost margins have to be considered reduced-form parameters (amixtures of supply and demand side coefficients) when using common pricedeflators as in model (S6).

All the results presented in Table 1 and 2 are based on models specifiedin such a way that deviation from constant return to scale are measuredexplicitly. Therefore, the well-known problem of small capital coefficient issomehow hidden. Nevertheless, it is simple to notice that the capital co-efficients take low values across all the specifications, thus confirming theproblem found with estimators in differences.32 As mentioned in Section 2,these poor results are possibly due to the fact that the available servicesfrom this quasi-fixed input are not fully utilised in all the faces of the busi-ness cycle. Table 3 presents the results for specification (S7). The capitalcoefficient for firms with high degree of capacity utilisation, αh3 , is positive,large and statistically different from zero. On the contrary, the estimated co-efficient for firms with low factor usage, αl3, takes a negative sign although itis not statistically different from zero. Results reported in Table 4 strengthenthe findings above. Using the “short-run” capital C* as instrument, we gethigher and more reasonable estimates of capital coefficient. The point esti-mate of α3 in column 1 of Table 4 is more than three times higher than thelower value of 0.02 that can be inferred from column 1 of Table 1.

INSERT TABLE 3 and 4 ABOUT HERE

As the capacity utilization is not a variable usually provided in manydata set, in Appendix D I show that similar results can be obtained usingother proxies, such as the individual firm productivity (measured as outputper employee) or the sample growth rates of production. The results pre-sented above are consistent with our claim that the low and often insignifi-cant capital coefficients usually found in productivity analysis are due to the31This value is slightly higher than -4.31 as reported above but it can be considered in

the same order of magnitude.32For example, the estimated values of the capital coefficient, α3, are around 0.02 and

0.05 in column 1 and 4 of Table 1, respectively.

20

existence of short-run equilibrium characterised by sub-optimal utilisation ofthis quasi-fixed input.

5 Conclusions

This paper explores the reasons why GMM estimators of production func-tion parameters are generally found to produce unsatisfactory results: lowand often insignificant capital coefficient and unreasonably low estimates ofreturns to scale. I attribute this well-known finding to the inaccurate con-struction of the variables used in production function analysis. In particular,I suggest that the problem lies in the use of common price deflators as wellas in a capital stock that does not reflect the time flow of the services. I showthat the practice of using deflated sales and deflated expenditure in materialsinstead of a “quantity” measure of output and intermediate inputs leads tolower scale estimates, mainly due to a relevant downward bias in the labourcoefficient. At the same time, the unavailability of firm-specific price dataintroduces a large upward bias in estimating the elasticity of output withrespect to technological innovation. I propose a simple test to assess theeffect of industry wide price deflators based on comparing the two mark-upparameters of specification (S5). Moreover, I show that the KG suggestion ofusing qIt to infer the true values of input coefficient seems to work adequately.

As far as the capital variable is concerned, a dramatic improvement in co-efficient estimates is obtained when we take into account the annual variationsin capacity utilisation. As described by Doms and Dune (1998), investmentis largely a lumpy activity characterised by period of low investment followedby bursts of investments.33 When we specify our production function in lev-els, the impossibility of full capital utilisation in a year of intense investmentactivity does not trigger the estimate of capital coefficients as long as firmsmove towards their long-run equilibrium during the sample period. On thecontrary, going to first-differences can reveal how dramatic can be the incor-rect definition of capital inputs since there are a few periods of intense capitalgrowth but these do not necessary coincide with proportional variations in

33This finding is confirmed for the case of Spain. Sanchez (2002) finds evidences ofinvestment spikes using the same survey (ESEE) of this study.

21

capital services if an unexpected under utilisation of capacity occurs. I showthat if the flow of capital services is adequately measured, there is a notableimprovement in the estimation.

An important message that we can get from the set of results reportedabove is that imposing the constant return to scale restriction even when wefind lower estimates of returns to scale is not incorrect. Measurement errorsof output and inputs can seriously bias downward the estimates of inputcoefficients and the constant return hypothesis would not be rejected if itwere not for these measurement errors. Blundell and Bond (2000) find thatmore reasonable parameters estimates can be obtained using a composedor extended GMM estimators, where lagged first differences are consideredinformative instruments for the endogenous variables in levels. This papershows that the greatest improvements in estimating the production func-tion parameters can be derived from a better construction of the (dependentand independent) variables more than from a refinement of the econometrictechniques. Although this requires the use of detailed data that are usuallynot available, I show that more reliable results can be obtained using otherproxies, in particular for the degree of capacity utilisation.

22

Appendix A: Data and VariableA1 Construction of Data SampleThe survey provides data on manufacturing firms with 10 or more em-

ployees. When this was designed, all firms with more than 200 employeeswere required to participate while a representative sample of about 5% of thefirms with 200 or less employees was randomly selected. In 1990, the firstyear of the panel, 715 firms with more than 200 employees were surveyed,which accounts for 68% of all the Spanish firms of this size. Newly estab-lished firms have been added every subsequent year to replace the exits dueto death and attrition. We start with a sample of 3,151 firms in an unbal-anced panel data. The total number of observations is 18,680. We then cleanour dataset according to the following criteria:

1) We remove all the observations where the log difference of the R&Dvariable between two consecutive years exceeds 4 in absolute value. Thisremoves 64 observations.2) We drop the observations where the log difference of the capital stock

variable between two consecutive years exceed 2 in absolute value. Thisremoves other 81 observations.The subsample we use in our study consists of all the firms that have

been surveyed for at least three years. There are 2,430 firms satisfying thiscondition, for a total number of 16,637 observations. At this point, we re-move any observations for which the data required to the estimation are notavailable. In the tables showing the results of the estimation, we report theexact number of observations making up the final samples.

The original industrial classification reported in the survey is based on 18sectors. A classification based on 15 sectors is used here since deflators forgross production and were found at this level because of aggregation. These15 sectors are:

1) Ferrous and non ferrous metals; 2) Non-metallic minerals; 3) Chemicalproducts; 4) Metal products; 5) Industrial and agricultural machinery; 6)Office and data processing machine; 7) Electrical and electronic goods; 8)Vehicles, cars and motors; 9) Other transport equipment; 10) Food and bev-erages; 11) Textiles, clothing and shoes; 12) Timber and furniture; 13) Paperand printing; 14) Rubber and plastic products; 15) Other manufacturingproducts.

23

A2 Description of VariablesCapacity Utilization (U): Yearly average rate of capacity utilization re-

ported by the firmsDeflated Gross Production (Y): Gross production is defined as the sum of

sales and the variation of inventories. We deflate the nominal amount usingoutput deflators at fifteen-industry level.Deflated Expenditure in Materials (N): We deflate total expenditure in

intermediate inputs as reported in the data set using a deflator for non-durable goods.Deflated Value Added (deVA): It has been computed subtracting N from

the gross production Y (see above the two definitions).Employment (EMP): Approximation to the average number of works dur-

ing the year; it does not consider employees engaged in R&D activities.Labour (L): Labour consists of the total hours of work. It has been

constructed using the number of works, adjusted for the double counting ofR&D employees, times the normal hours plus overtime and minus lost hours.Materials (M): Nominal materials are given by the sum of purchases and

external services minus the variation of intermediate inventories. We usefirms’ specific deflator based on the variation in the cost of raw materialsand energy as reported by the firm.R&D stocks (R): This variable is constructed using the perpetual inven-

tory method, assuming a depreciation rate of 15%, ρ = 0.15. Computationis fully explained in Section 3.Output (Q): Nominal output is defined as the sum of sales and the varia-

tion of inventories. We deflate the nominal amount using the firms’s specificoutput price as reported in the data set.Physical Capital (C): It has been constructed capitalising firms’ invest-

ments in machinery and equipment and using sectorial rates of depreciation.The capital stock does not include buildings. This variable is taken fromMartin and Suarez (1997).Value Added (VA): It has been computed subtracting the value of ma-

terials, M, from output, Q. As both these variables are based on firm-levelprices, we consider VA as our “true” measure of production.

24

Growth rates of the variablea; sample period: 1990-1999.Variable Name Mean St. Dev.Output Q 0.0369 0.279Deflated Gross Prod. Y 0.0282 0.279Value Added V A 0.0505 0.463Deflated Value Added deV A 0.0018 0.502Labour L 0.0015 0.211Employment EMP 0.0023 0.194Materials M 0.0266 0.382Deflated Expen. Mater. N 0.0276 0.383Physical Capital C 0.0775 0.248Capacity Utilizationb U 0.80 0.156R&D stocks R 0.0366 0.232Note:aWe use the logarithmic differences of the variables.bVariable expressed in Levels

Appendix B: Decomposing the “omitted price”biasResults reported in this table show the impact on production function

parameters of the term (pit − pIt) and (git − gIt), separately. Column 1reports estimation results using “true” measure of output, Q, and materials,M . In column2, deflated gross production, Y , is used as dependent variableand materials, M , as a regressor (this allows us to isolate the bias due to(pit − pIt)). In column 3, I use deflated expenditure in intermediate inputs,N , and output, Q, as dependent variable (this allows us to isolate the bias dueto (git− gIt)). Finally, column 4 reports the overall effect on the estimation ofthe parameter vector α when using industry-wide output and input deflators.Notice that results shown in column1 and 4 are equivalent to those reportedin Table 1.

25

Estimation method: GMM estimatesaDep.Vbls Q Y Q YModel 1 2 3 4

Labour 0.362*** 0.344*** 0.318*** 0.296***(0.063) (0.061) (0.065) (0.062)

Materials 0.541*** 0.537*** 0.565*** 0.570***(0.049) (0.049) (0.052) (0.051)

Capital -0.077 -0.091* -0.101* -0.110**(0.056) (0.055) (0.056) (0.054)

R&D 0.112** 0.143*** 0.166*** 0.192***(0.055) (0.053) (0.054) (0.050)

Cap.Util 0.286*** 0.288*** 0.241*** 0.238***(0.089) (0.088) (0.091) (0.088)

Ind. Dum. Inc. Inc. Inc. Inc.Time Dum. Inc. Inc. Inc. Inc.Period 1990-99 1990-99 1990-99 1990-99N. Obs 11,476 11,476 11,476 11,476Sargan T. 90.48 98.5 87.68 88.86(df) (82) (82) (82) (82)m1 -10.45 -10.54 -10.34 -10.56m2 0.01 -0.61 -0.16 -0.66Heteroskedasticity robust standard errors shown in parentheses.* significant at 10% level; ** significant at 5%; *** significant at 1%.aIV s: number of workers, physical capital and capacity utilization lagged

levels from t− 2 to t− 4; materials lagged levels from t−2 to t− 3 and R&Dcapital lagged levels in t − 2; (exogenous variables) growth of capital andgrowth in industry output (for specification in column 3).

Appendix C: Measurement Errors of Physi-cal CapitalIn this Appendix I show that the bias of capital coefficients is likely to

be amplified when we specify the production function in differences. Thisanalysis mimics the one presented in Chapter 4 of Arellano (2002), to whichwe refer for detailed explanations. Consider first the model in levels yit =αx†it + uit and suppose that the regressor is measured with an error, so thatwe actually observe: xit = x

†it+εit (i). Under this framework, the existence of

temporary equilibria characterized by sub-optimal utilisation of the available

26

physical capital can be modelled as a measurement error εit = (1 − θit)xitwhere θit is the proportion of capital stock xit that is actually used. Then,we can rewrite (i) as xit = x

†it + (1− θit)xit or, rearranging, xit = (1/θit)x

†it.

Now, the point estimate of the capital coefficient using xit will be given by:

α = cov(xit,yit)var(xit)

=cov((1/θit)x

†it,yit)

var((1/θit)x†it)

=(1/θit)cov(x

†it,yit)

(1/θit)2var(x†it)= θit ∗ α (ii)

where α is the true value of the capital coefficient if we use x†it. In mydataset, firms use in the average 80% of their capacity utilisation, θ = 0.8,so that we can expect a point estimate of the capital coefficient that is lowerthan its true value. Now, I analyse the effect of measurement errors whenthe model is specified in first differences: ∆yit = α∆x†it + ∆uit. The esti-

mated coefficient is given by: cov(∆xit,∆yit)var(∆xit)

=cov((1/θit)x

†it−(1/θit−1)x†it−1,∆yit)

var((1/θit)x†it−(1/θit−1)x†it−1)

=

cov( 1θitx†it− 1

θitx†it−1+

1θitx†it−1− 1

θit−1 x†it−1,∆yit)

var( 1θitx†it− 1

θitx†it−1+

1θitx†it−1− 1

θit−1 x†it−1)

=cov( 1

θit∆x†it+(

1θit− 1

θit−1 )x†it−1,∆yit)

var(( 1θit)∆x†it+(

1θit− 1

θit−1 )x†it−1)

=cov( 1

θit∆x†it,∆yit)+cov((

1θit− 1

θit−1 )x†it−1,∆yit)

var( 1θit

∆x†it)+var((1θit− 1

θit−1 )x†it−1)+2cov(

1θit

∆x†it,(1θit− 1

θit−1 )x†it−1)

(iii).

If we assume that the covariance term in the denominator and the secondterm in the numerator are close to zero (given that most of the series ofoutput and inputs are persistent, first differences have low correlation withpast levels as noticed by Blundell and Bond (2000)), we can rewrite (iii) (afterdividing by var( 1

θit∆x†it) the denominator and the numerator) as

θit∗α1+λ

(iv)

where λ =var(( 1

θit− 1

θit−1 )x†it−1)

var( 1θit

∆x†it). Comparing (ii) and (iv), we find that the bias

of the capital coefficient for first difference equations is higher than the onefor levels. In order to minimize this bias we need to find two (consecutive)values of θit and θit−1 that are both close to 1. In this way, the downwardbias in the numerator (as θit will be close to one) and in the denominator (asthe term ( 1

θit− 1

θit−1) will be close to zero) is simultaneously reduced

Appendix D: Temporary EquilibriumAs data on capacity utilization are usually not available, this Appendix

shows that similar results to those presented in Table 3 can be obtained usingother proxies, namely the individual firm productivity and the “sample”growth rates of production. Let’s start considering the first of these twovariables.After computing the worker productivity, prod, measured as output per

employee, I determine the maximum value,maxprod, and the minimum value,

27

minprod, of this variable for each firm across all the years surveyed and, then,the following rule is applied to infer the possible capacity utilization:CapUtil2it = (100− 50 ∗ (maxprod− prodit)/(maxprod−minprod))The variable CapUtil2 takes values between 50 (in the year of the lowest

worker productivity) and 100 (in the year of highest productivity). Thenusing this variable, I determine two distinct capital coefficients as in specifi-cation (S7) of Section 3.2:qit = α1lit + α2mit + αh3D1cit + αl3D2cit + u

pit (S7b)

where D1 is a dummy that takes value 1 if the variable CapUtil2 is greaterthan 80% in both year t and t+1 while D2 takes value 1 if this condition isnot satisfied. Column 1 of Table D1 reports the results obtained using thisapproach.The second variable used as a proxy for capacity utilization is the “sam-

ple” growth of production. Once computed the average growth in each yearof the survey, I consider the 4 years where the growth rate records a greaterincrease (or a smaller decrease) compared to the previous year and we esti-mate the capital coefficient for these 4 years separately from the other years.This means that the dummy D1 in equation (S7b) takes value one for all theobservations in these 4 years (namely, 1991, 1994, 1997 and 1998) and D2

takes value 1 for the observations in the remaining years. Results reportedin column 2 show that even under this simple classification, rather differentpoint estimate of the elasticity of output with respect to capital are obtained.

Estimation method: GMM estimatesDep.Vbls Q Q

1a 2a

Labour 0.313*** (0.075) 0.274*** (0.073)Materials 0.623*** (0.045) 0.613*** (0.046)D1 ∗ Capital 0.155* (0.085) 0.095** (0.038)D2 ∗ Capital -0.021 (0.029) -0.048 (0.035)Ind. Dum. Inc. Inc.Time Dum. Inc. Inc.Period 1990-99 1990-97N. Obs 11,581 11,581Sargan T. (df) 56.83 (54) 54.8 (54)m1 -9.46 -9.34m2 -0.09 -0.21* significant at 10% level; ** significant at 5%; *** significant at 1%.

28

aIV s: number of workers and physical capital lagged levels from t− 2 tot−4; materials lagged levels from t−2 to t−3; (exogenous variables) growthof capital.

29

References[1] Arellano, M. and Bond S. (1991), “Some Test of Specification for Panel

Data: Monte Carlo Evidence and an Application to Employment Equa-tions” Review of Economic Studies, Vol. 58, pp. 277-297.

[2] Beaulieu, J. and Mattey J. (1999), “The Effects of General Inflationand Idiosyncratic Cost Shocks on Within-comodity Price Dispertion:Evidence from Microdata” Review of Economics and Statistics, Vol. 81,pp. 205-216.

[3] Berndt, E.R. and Fuss M. A. (1986), “Productivity Measurement withAdjustment for Variations in Capacity Utilization and Other Forms ofTemporary Equilibrium” Journal of Economietrics, Vol. 33, pp. 7-29.

[4] Blundell, R. and Bond S.R. (2000), “GMM Estimation with PersistentPanel Data: An Application to Production Functions”, EconometricReviews, Vol. 19, pp. 321-340.

[5] Cooper, R., Haltiwanger J. and Power L. (1999) “Machine Replacementand the Business Cycle: Lumps and Bumps” American Economic Re-view, Vol. 89, pp. 921-946.

[6] Doms, M and Dunne T. (1998), “Capital Adjustment Patterns in Man-ufacturing Plants” Review of Economic Dynamics, Vol. 1, pp. 409-429.

[7] Garcia, A., Jaumandreu, J. and Rodriguez, C. (2002), “Innovation andjobs: evidence from manufacturing firms”. mimeo.

[8] Griliches, Z. (1979), “Issues in Assessing the Contribution of R&D toProductivity Growth”. Bell Journal of Economics, Vol. 10, pp. 92-116.

[9] Griliches, Z. and Mairesse J. (1995), “Production Function: The Searchfor Indentification”. NBER Working Paper, No. 5067.

[10] Hall, B.H and Mairesse J. (1995), “Exploring the Relationship be-tween R&D and Productivity in French Manufactuing Firms”. Journalof Economietrics, Vol. 65, pp. 263-293.

[11] Hall, R.E. (1988), “The Relation between Price and Marginal Cost inU.S. Industry” Journal of Political Economy, Vol. 96, pp. 921-947.

30

[12] Kettle, T.J. (1999), “Market Power, Scale Economies and Productiv-ity: Estimates from a Panel of Establishment Data”. The Journal ofIndustrial Economics, Vol. 48, pp. 451-476.

[13] Kettle, T.J. (1996), “R&D, Scope Economies, and Plant Performance”.RAND Journal of Economics, Vol. 27, pp. 502-522.

[14] Kettle, T.J. and Griliches Z (1996), “The inconsistency of CommonScale Estimators when Output Prices are Unobserved and Endogenous”.Journal of Applied Econometrics, Vol. 11, pp. 343-361.

[15] Martin, A. and Suarez C. (1997), “El Stock de Capital para las Em-presas de la Encuesta sobre Estrategias Empresariales”. Documento In-terno n.13, Programa de Investigaciones Económicas, Fundación Em-presa Pública

[16] Ornaghi C. (2002) “Spillovers in Product and Process Innovation: Ev-idence from Spain”,Working Papers 02-32, Universidad Carlos III deMadrid..

[17] Sanchez R. (2002) “Estimation of a Dynamic Discrete Choice Model ofIrreversible Investment”,Working Papers 01-56, Universidad Carlos IIIde Madrid.

31

TABLES

Table 1. Production Fuction EstimatesEstimation method: GMM estimatesDep.Vbls Q Y Y V A deV AModel (S1)a (S2)a (S3)a (S1)b (S2)b

Labour 0.362*** 0.296*** 0.296*** 0.775*** 0.621***(0.063) (0.062) (0.062) (0.125) (0.137)

Materials 0.541*** 0.570*** 0563*** - -(0.049) (0.051) (0.051)

Capital -0.077 -0.110** -0.117** -0.177 -0.293**(0.056) (0.054) (0.055) (0.112) (0.122)

R&D 0.112** 0.192*** 0.190*** 0.268*** 0.441***(0.055) (0.050) (0.050) (0.104) (0.108)

Cap.Util 0.286*** 0.238*** 0.257*** 0.536*** 0.688***(0.089) (0.088) (0.089) (0.178) (0.206)

Ind.Output - - 0.232*** - -(0.040)

Ind. Dum. Inc. Inc. Inc. Inc. Inc.Time Dum. Inc. Inc. Inc. Inc. Inc.Period 1990-99 1990-99 1990-99 1990-97 1990-97N. Obs 11,476 11,476 11,476 8,687 8,687Sargan T. 90.48 88.86 94.70 57.44 59.28(df) (82) (82) (82) (66) (66)m1 -10.45 -10.56 -10.55 -10.45 -10.83m2 0.01 -0.66 -0.56 -1.09 -1.64Heteroskedasticity robust standard errors shown in parentheses.* significant at 10% level; ** significatn at 5%; *** significant at 1%.aIV s: number of workers, physical capital and capacity utilisation lagged

levels from t − 2 to t − 4; materials lagged levels from t − 2 to t − 3 andR&D capital lagged levels in t − 2; (exogenous variables) growth of capitaland growth in industry output (for specification in column 3).

bIV s: number of workers and capacity utilisation lagged levels from t− 2to t−4; physical capital from t to t−4 (as it is considered exgoneous); R&Dcapital lagged levels from t− 2 to t− 3;

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Table 2. Mark-up EstimatesEstimation method: GMM estimatesDep.Vbls Q Y YModel (S4)a (S5)a (S6)a

Labour 1.095*** 0.929*** 0.926***(0.113) (0.118) (0.118)

Materials 1.082*** 1.104*** 1.104***(0.061) (0.056) (0.056)

Capital -0.072 -0.105** -0.106**(0.048) (0.047) (0.048)

R&D 0.070** 0.143*** 0.143***(0.035) (0.033) (0.034)

Cap.Util 0.028** 0.039*** 0.039***(0.012) (0.012) (0.012)

Ind.Output - - 0.165***(0.036)

Ind. Dum. Inc. Inc. Inc.Time Dum. Inc. Inc. Inc.Period 1990-99 1990-99 1990-99N. Obs 9,969 9,969 9,969Sargan T. 82.64 78.56 81.65(df) (78) (78) (78)m1 -12.71 -13.64 -13.59m2 -1.08 -2.41 -2.26* significant at 10% level; ** significatn at 5%; *** significant at 1%.aIV s: number of workers from t − 2 to t − 5; physical capital from t to

t − 3 (as it is considered exgoneous); materials lagged levels from t − 2 tot − 3 and R&D capital lagged levels from t− 2; (exogenous) growth rate ofcapacity utilisation.

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Table 3. Temporary Equilibrium 1Estimation method: GMM estimatesDep.Vbls Q VAModel (S7)a (S7)b

Labour 0.374*** 0.767***(0.064) (0.127)

Materials 0.516*** -(0.049)

D1 ∗ Capital 0.181** 0.284*(0.091) (0.162)

D2 ∗ Capital -0.078 -0.044(0.059) (0.073)

Cap.Util 0.326*** 0.622***(0.095) (0.201)

R&D 0.122** 0.251**(0.056) (0.106)

Ind. Dum. Inc. Inc.Time Dum. Inc. Inc.Period 1990-99 1990-97N. Obs 11,476 8,687Sargan T. 83.18 53.12(df) (81) (65)m1 -10.26 -10.41m2 0.01 -1.01* significant at 10% level; ** significatn at 5%; *** significant at 1%.aIV s: number of workers, physical capital and capacity utilisation lagged

levels from t− 2 to t− 4; materials lagged levels from t−2 to t− 3 and R&Dcapital lagged levels in t− 2; (exogenous variables) growth of capital.

bIV s: number of workers and capacity utilisation lagged levels from t− 2to t−4; physical capital from t to t−4 (as it is considered exgoneous); R&Dcapital lagged levels from t− 2 to t− 3;

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Table 4. Temporary Equilibrium 2Estimation method: GMM estimatesDep.Vbls Q VAModel (S8)a (S8)b

Const. 0.00 0.024(0.011) (0.027)

Labour 0.347*** 0.794***(0.069) (0.150)

Materials 0.533*** -(0.059)

Capital 0.076*** 0.123***(0.025) (0.041)

R&D 0.134** 0.296**(0.058) (0.155)

Ind. Dum. Inc. Inc.Time Dum. Inc. Inc.Period 1990-99 1990-97N. Obs 11,476 8,687Sargan T. 82.97 35.71(df) (67) (39)m1 -8.69 -10.25m2 -1.24 -1.74* significant at 10% level; ** significatn at 5%; *** significant at 1%.aIV s: number of workers lagged levels from t − 2 to t − 5; capacity

utilisation from t− 2 to t− 4 and materials from t− 2 to t− 3; R&D capitallagged levels in t−2; growth of “short-run” capital, C*, is used as instrumentfor stock of capital.

bIV s: number of workers and capacity utilisation lagged levels from t− 2to t− 4; R&D capital lagged levels from t− 2 to t− 3; growth of “short-run”capital, C*, is used as instrument for stock of capital.

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